Let us see if this is true
No
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Let
Proof.
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The bigger problem is the claim that the area of the sector is
The next image is also supposed to be svg, but is tikz :(
There are a couple of small and a large problem with the above proof that are generally not addressed in most books.
The fact that the area of
Similarly although a bit less of an issue is that no argument is generally given for showing that the area of the sector
The continuity of
The bigger problem is the claim that the area of the sector is
But how do we know that the area of the circle is
a. A standard proof goes as follows.
The circle of radius
But here we have used that the derivative of
b. There is also another proof that tries to avoid differentiating any trigonometric function. This is the so-called shell method, where we divide the circle into very small rings (infinitesimally small) of radius
c. How about a non-integral proof? Like say Archimedes' proof, where he shows that the area of the circle is neither greater than nor less than the area of a triangle with base equal to the circumference and height equal to the radius?
If one reads very carefully, the proof assumes that if a curve converges pointwise to another curve (in this case, inscribed polygons converging to a circle), then so does the areas. This is not rigorously justified and can fall prey to the staircase paradox. The rigorous justification again uses the limit in question, so... we get the point.
We need to answer each of these questions if we are to make sure the calculation of the limit above is airtight. We proceed to do exactly that.
Is the above getting rendered? The commutative diagram I mean.